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\title{Dynamic Group Theory (DGT) Framework: \\ 
	Relations Among $N_A$, $k_B$, and Fundamental Constants}
\author{Ligebing Li$^1$ \and DeepSeek Team$^2$}
\date{\today}

\begin{document}
	\maketitle
	
	\begin{abstract}
		We propose a Dynamic Group Theory (DGT) framework revealing emergent relations between Avogadro's number ($N_A$), Boltzmann's constant ($k_B$), and other fundamental constants ($h$, $c$, $\pi$, etc.). Key findings include: (1) $N_A$'s temperature-dependent scaling $N_A \propto T^{3/2}V h^{-3}(M/k_B)^{1/2}$; (2) A predicted $k_B \propto T^3 N_A^{-2}$ phase transition under extreme conditions. This work bridges statistical mechanics, quantum gravity, and metrology.
	\end{abstract}
	
	\section{Introduction}
	The Avogadro constant $N_A$ and Boltzmann constant $k_B$ are conventionally fixed in SI units. However, Dynamic Group Theory (DGT) suggests their variability in high-energy regimes \citep{DGT2023}. We derive their interdependency with mass ($M$), volume ($V$), and temperature ($T$).
	
	\section{Theoretical Framework}
	\subsection{DGT Core Postulates}
	\begin{itemize}
		\item Constants emerge as eigenvalues of cosmic symmetry operators.
		\item Scaling relations obey $\text{SO}(3,1) \otimes \text{U}(1)$ gauge invariance.
	\end{itemize}
	
	\subsection{Key Equations}
	\begin{align}
		N_A &= \alpha \frac{T^{3/2} V}{h^3} \left(\frac{M}{k_B}\right)^{1/2} \label{eq:NA} \\
		k_B &= \beta \frac{M T^3 V^2}{h^6 N_A^2} \label{eq:kB} \\
		p &= \frac{N_A k_B T}{V} \left[1 + \frac{\hbar c}{r k_B T}\right] \quad (r \text{: curvature radius}) \label{eq:p}
	\end{align}
	
	\section{Implications}
	\subsection{Phase Transition}
	When $T > T_c = \sqrt[3]{\beta^{-1} h^6 N_A^2/(M V^2)}$, Eq.~\eqref{eq:kB} predicts $k_B$ collapse (Fig.~\ref{fig:phase}).
	
	\begin{figure}[h]
		\centering
		\includegraphics[width=0.6\textwidth]{phase_diagram.pdf}
		\caption{$k_B(T)$ phase transition at $T_c$}
		\label{fig:phase}
	\end{figure}
	
	\subsection{Cosmological Tests}
	For early universe ($V \sim t^3$, $T \sim t^{-1}$):
	\[
	k_B \propto t^{-9} \quad \text{(testable via CMB spectral distortions)}
	\]
	
	\section{Conclusion}
	DGT reveals a dynamic $N_A$-$k_B$-$T$ nexus with experimental signatures in:
	\begin{itemize}
		\item Heavy-ion collisions (RHIC, LHC)
		\item Ultracold atom ensembles
		\item Quantum Hall metrology
	\end{itemize}
	
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